Core-Level Photoelectron Angular Distributions at the Liquid–Vapor Interface

Conspectus Photoelectron spectroscopy (PES) is a powerful tool for the investigation of liquid–vapor interfaces, with applications in many fields from environmental chemistry to fundamental physics. Among the aspects that have been addressed with PES is the question of how molecules and ions arrange and distribute themselves within the interface, that is, the first few nanometers into solution. This information is of crucial importance, for instance, for atmospheric chemistry, to determine which species are exposed in what concentration to the gas-phase environment. Other topics of interest include the surface propensity of surfactants, their tendency for orientation and self-assembly, as well as ion double layers beneath the liquid–vapor interface. The chemical specificity and surface sensitivity of PES make it in principle well suited for this endeavor. Ideally, one would want to access complete atomic-density distributions along the surface normal, which, however, is difficult to achieve experimentally for reasons to be outlined in this Account. A major complication is the lack of accurate information on electron transport and scattering properties, especially in the kinetic-energy regime below 100 eV, a pre-requisite to retrieving the depth information contained in photoelectron signals. In this Account, we discuss the measurement of the photoelectron angular distributions (PADs) as a way to obtain depth information. Photoelectrons scatter with a certain probability when moving through the bulk liquid before being expelled into a vacuum. Elastic scattering changes the electron direction without a change in the electron kinetic energy, in contrast to inelastic scattering. Random elastic-scattering events usually lead to a reduction of the measured anisotropy as compared to the initial, that is, nascent PAD. This effect that would be considered parasitic when attempting to retrieve information on photoionization dynamics from nascent liquid-phase PADs can be turned into a powerful tool to access information on elastic scattering, and hence probing depth, by measuring core-level PADs. Core-level PADs are relatively unaffected by effects other than elastic scattering, such as orbital character changes due to solvation. By comparing a molecule’s gas-phase angular anisotropy, assumed to represent the nascent PAD, with its liquid-phase anisotropy, one can estimate the magnitude of elastic versus inelastic scattering experienced by photoelectrons on their way to the surface from the site at which they were generated. Scattering events increase with increasing depth into solution, and thus it is possible to correlate the observed reduction in angular anisotropy with the depth below the surface along the surface normal. We will showcase this approach for a few examples. In particular, our recent works on surfactant molecules demonstrated that one can indeed probe atomic distances within these molecules with a high sensitivity of ∼1 Å resolution along the surface normal. We were also able to show that the anisotropy reduction scales linearly with the distance along the surface normal within certain limits. The limits and prospects of this technique are discussed at the end, with a focus on possible future applications, including depth profiling at solid–vapor interfaces.


Experimental Details
The measurements rely on comparing absolute photoelectron intensities for successively measured spectra at different polarization angles. This requires particularly stable conditions. Photon-flux variations (including differences between different polarization angles) need to be measured reliably. This is typically done using a calibrated photodiode ahead of the experimental chamber, or by monitoring the mirror current of the last mirror of the beamline. Typically, drifts in the intensity due to an unstable liquid-jet or beam-focus position are monitored by regularly re-measuring at one reference polarization angle (usually 0 • as it is the most intense one). The measurements are also affected by the necessarily non-perfect measurement conditions. To discuss the various sources of imperfections, we first introduce a modified version of Eq. 1 from the main text that is effectively used for fitting data: (1 + 3p cos(2(θ − θ 0 ))) (S1) Here, θ 0 is an offset of the polarization angle and p is the degree of linear polarization (Stokes parameter) of the light. This equation is equivalent to Eq. 1 of the main text for p = 1 and θ 0 = 0. An offset θ 0 of the polarization angle can stem from an actual offset of the true polarization direction relative to the nominal one, originating, e.g., from a small mechanical offset of the undulator, from to slight deviations from the experimental geometry sketched in the main text (i.e., alignment of the detector relative to the light propagation axis), or even possibly stray magnetic fields in the chamber that bend electron trajectories. θ 0 can be determined from the fit. The degree of linear polarization of the light p, on the other hand, stems from a non-ideal linear (i.e., slightly elliptic) polarization, but cannot be determined from fitting because the p parameter is quasi-degenerate with the β parameter. On the other hand, that means that if p is different from 1, this will result in an overall reduction of the fitted β values compared with the true ones. If one only wishes to compare relative β values, which is our case here, this is of little importance.
Another factor is the non-zero angular acceptance of the detector, meaning a measurement at a certain angle actually averages over an angular range. This effectively lowers all the measured values of β uniformly, just like depolarization effects. One can theoretically calculate what would be the effect of a non-zero circular angular acceptance on an initial PAD with anisotropy parameter β. The modified distribution reads: where θ a is the half-acceptance angle. Thus, for a half-angular acceptance of, e.g., ±12 • , the measured value of β is reduced by 5% compared to the initial one. This is well corroborated by the PAD simulations described below; changing the analyzer acceptance angle in the SESSA simulation (see below) leads to a reduction of simulated β values in agreement with Eq. S2. However, the stated angular acceptance of commercial analyzers is only in the non-dispersive direction. Angular acceptance in the dispersive direction is most likely smaller and quite different, and depends on the analyzer settings (e.g., the entrance slits). Thus, the actual effect of angular acceptance would also have to be determined empirically.
For the same reasons mentioned above, we ignore the angular-acceptance effect here. In principle, the empirical determination of total depolarization effects (angular acceptance plus depolarization of the beam) would be possible by, e.g., measuring the PAD of an s orbital of a rare gas, for which β = 2 at all photoelectron kinetic energies. In practice, because of the combination of these two different effects, depolarization is likely to depend on both the photon energy (beam depolarization) and on the electron kinetic energy (analyzer settings), which makes it little practical to determine. The measurement of β = 2 at high eKE for gas-phase H 2 O in key reference 1 suggests that depolarization effects are small, at least in this specific study.

Analytical approximation of the DCS
In key reference 1, the DCS was approximated by a Gaussian: with a characteristic angle ϕ = 17 • extrapolated from gas-phase water data. For a Gaussian DCS, it is possible to calculate analytically I * (θ) from Eq. 2 in the main text and to identify the modified anisotropy β * : Despite its simplicity, this analytical model is relatively robust since Eq. S4 turns out to be valid for non-Gaussian DCS. Calculating the n-fold convolution of a nascent PAD with various DCS and deriving the resulting value of β yields results that can be effectively fitted with Eq. S4, with ϕ as a free parameter. This is shown in the supplementary information of key reference 3.
For n → 0, the expression reduces to: SESSA allows to perform simple simulations that provide an intuitive understanding of how the measured β parameter varies as a function of several parameters, especially the IMFP and EMFP. In Fig. S1, we simulated a solid water model (modeled by a homogeneous 2:1 H:O solid sample) and calculated the β value as a function of EMFP, also explicitly varying the IMFP and eKE. The initial β was always 2. The blue points correspond to calculations where we set EMFP = IMFP, i.e., a constant elastic-to-inelastic ratio. This means a constant average number of collisions for the photoelectrons, leading as expected to a constant value of β, independent of the absolute value of the EMFP. EMFP = IMFP is a situation close to what would be roughly expected for water for eKE ∼100-150 eV and higher, which in turn does lead to a constant R β [3]. The red and black points in Fig. S1 correspond to calculations where we set IMFP = 10 (a.u.) and varied the value of EMFP, effectively varying the EMFP-to-IMFP ratio. We observe a characteristic curve shape, reminiscent of the one obtained in Fig. 1 of the main text, but which, however, should not be confused with it, since when eKE is varied also the IMFP and DCS will change. The red and black points correspond to eKE = 100 eV (black) and 500 eV (red), respectively, the only difference between the two eKEs being the shape of the DCS (since EMFP and IMFP are explicitly fixed), with the DCS at 100 eV containing a significant backscattering component, thus enhancing β reduction. Simulations in a cylindrical geometry, performed in the manner of Ref. [1], showed no significant difference with the plane geometry.

SESSA simulations
In Fig. S2, we simulated PADs and calculated their β parameters for different values of the nascent β, which is another parameter that can be freely changed in the software. The results show a perfectly linear correlation.